# Coverage for pygeodesy/ellipsoidalVincenty.py : 99%

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Pure Python implementation of geodesy tools for ellipsoidal earth models, transcribed from JavaScript originals by I{(C) Chris Veness 2005-2016} and published under the same MIT Licence**, see U{Vincenty geodesics <https://www.Movable-Type.co.UK/scripts/LatLongVincenty.html>}. More at U{GeographicLib<https://PyPI.org/project/geographiclib>} and U{GeoPy<https://PyPI.org/project/geopy>}. Calculate geodesic distance between two points using the U{Vincenty <https://WikiPedia.org/wiki/Vincenty's_formulae>} formulae and one of several ellipsoidal earth models. The default model is WGS-84, the most accurate and widely used globally-applicable model for the earth ellipsoid. Other ellipsoids offering a better fit to the local geoid include Airy (1830) in the UK, Clarke (1880) in Africa, International 1924 in much of Europe, and GRS-67 in South America. North America (NAD83) and Australia (GDA) use GRS-80, which is equivalent to the WGS-84 model. Great-circle distance uses a spherical model of the earth with the mean earth radius defined by the International Union of Geodesy and Geophysics (IUGG) as M{(2 * a + b) / 3 = 6371008.7714150598} meter or approx. 6371009 meter (for WGS-84, resulting in an error of up to about 0.5%). Here's an example usage of C{ellipsoidalVincenty}: >>> from pygeodesy.ellipsoidalVincenty import LatLon >>> Newport_RI = LatLon(41.49008, -71.312796) >>> Cleveland_OH = LatLon(41.499498, -81.695391) >>> Newport_RI.distanceTo(Cleveland_OH) 866,455.4329158525 # meter You can change the ellipsoid model used by the Vincenty formulae as follows: >>> from pygeodesy import Datums >>> from pygeodesy.ellipsoidalVincenty import LatLon >>> p = LatLon(0, 0, datum=Datums.OSGB36) or by converting to anothor datum: >>> p = p.convertDatum(Datums.OSGB36) @newfield example: Example, Examples ''' # make sure int division yields float quotient from __future__ import division division = 1 / 2 # double check int division, see .datum.py if not division: raise ImportError('%s 1/2 == %d' % ('division', division)) del division from pygeodesy.basics import EPS, property_doc_, property_RO, _xkwds from pygeodesy.datum import Datums from pygeodesy.ecef import EcefVeness from pygeodesy.ellipsoidalBase import CartesianEllipsoidalBase, _intersect2, \ LatLonEllipsoidalBase, _TOL_M from pygeodesy.errors import _ValueError from pygeodesy.fmath import fpolynomial, hypot, hypot1 from pygeodesy.interns import _ambiguous_, NN, _no_convergence_ from pygeodesy.lazily import _ALL_LAZY, _ALL_OTHER from pygeodesy.named import Bearing2Tuple, Destination2Tuple, \ Distance3Tuple from pygeodesy.points import ispolar # PYCHOK exported from pygeodesy.units import Number_, Scalar_ from pygeodesy.utily import degrees90, degrees180, degrees360, \ sincos2, unroll180 from math import atan2, cos, radians, tan __all__ = _ALL_LAZY.ellipsoidalVincenty __version__ = '20.07.27' _antipodal_ = 'antipodal ' # _SPACE_ class VincentyError(_ValueError): '''Error raised from Vincenty's direct and inverse methods for coincident points or lack of convergence. ''' pass class Cartesian(CartesianEllipsoidalBase): '''Extended to convert geocentric, L{Cartesian} points to Vincenty-based, ellipsoidal, geodetic L{LatLon}. ''' def toLatLon(self, **LatLon_datum_kwds): # PYCHOK LatLon=LatLon, datum=None '''Convert this cartesian point to a C{Vincenty}-based geodetic point. @kwarg LatLon_datum_kwds: Optional L{LatLon}, B{C{datum}} and other keyword arguments, ignored if B{C{LatLon=None}}. Use B{C{LatLon=...}} to override this L{LatLon} class or specify B{C{LatLon=None}}. @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{LatLon}}, B{C{datum}} or other B{C{LatLon_datum_kwds}}. ''' kwds = _xkwds(LatLon_datum_kwds, LatLon=LatLon, datum=self.datum) return CartesianEllipsoidalBase.toLatLon(self, **kwds) class LatLon(LatLonEllipsoidalBase): '''Using the formulae devised by Thaddeus Vincenty (1975) with an ellipsoidal model of the earth to compute the geodesic distance and bearings between two given points or the destination point given an start point and initial bearing. Set the earth model to be used with the keyword argument datum. The default is Datums.WGS84, which is the most globally accurate. For other models, see the Datums in module datum. Note: This implementation of the Vincenty methods may not converge for some valid points, raising a VincentyError. In that case, a result may be obtained by increasing the epsilon and/or the iteration limit, see properties L{LatLon.epsilon} and L{LatLon.iterations}. ''' _Ecef = EcefVeness #: (INTERNAL) Preferred C{Ecef...} class, backward compatible. _epsilon = 1.0e-12 # about 0.006 mm _iteration = -1 # index, see C{.iteration} _iterations = 100 # vs Veness' 500 def bearingTo(self, other, wrap=False): # PYCHOK no cover '''DEPRECATED, use method C{initialBearingTo}. ''' return self.initialBearingTo(other, wrap=wrap) def bearingTo2(self, other, wrap=False): '''Compute the initial and final bearing (forward and reverse azimuth) from this to an other point, using Vincenty's inverse method. See methods L{initialBearingTo} and L{finalBearingTo} for more details. @arg other: The other point (L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: A L{Bearing2Tuple}C{(initial, final)}. @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit and/or if this and the B{C{other}} point are coincident or near-antipodal. ''' r = self._inverse(other, True, wrap) return self._xnamed(Bearing2Tuple(r.initial, r.final)) def destination(self, distance, bearing, height=None): '''Compute the destination point after having travelled for the given distance from this point along a geodesic given by an initial bearing, using Vincenty's direct method. See method L{destination2} for more details. @arg distance: Distance (C{meter}). @arg bearing: Initial bearing (compass C{degrees360}). @kwarg height: Optional height, overriding the default height (C{meter}, same units as C{distance}). @return: The destination point (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit. @example: >>> p = LatLon(-37.95103, 144.42487) >>> d = p.destination(54972.271, 306.86816) # 37.6528°S, 143.9265°E ''' r = self._direct(distance, bearing, True, height=height) return self._xnamed(r.destination) def destination2(self, distance, bearing, height=None): '''Compute the destination point and the final bearing (reverse azimuth) after having travelled for the given distance from this point along a geodesic given by an initial bearing, using Vincenty's direct method. The distance must be in the same units as this point's datum axes, conventionally C{meter}. The distance is measured on the surface of the ellipsoid, ignoring this point's height. The initial and final bearing (forward and reverse azimuth) are in compass C{degrees360}. The destination point's height and datum are set to this point's height and datum, unless the former is overridden. @arg distance: Distance (C{meter}). @arg bearing: Initial bearing (compass C{degrees360}). @kwarg height: Optional height, overriding the default height (C{meter}, same units as B{C{distance}}). @return: A L{Destination2Tuple}C{(destination, final)}. @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit. @example: >>> p = LatLon(-37.95103, 144.42487) >>> b = 306.86816 >>> d, f = p.destination2(54972.271, b) >>> d LatLon(37°39′10.14″S, 143°55′35.39″E) # 37.652818°S, 143.926498°E >>> f 307.1736313846706 ''' r = self._direct(distance, bearing, True, height=height) return self._xnamed(r) def distanceTo(self, other, wrap=False, **unused): # for -DistanceTo '''Compute the distance between this and an other point along a geodesic, using Vincenty's inverse method. See method L{distanceTo3} for more details. @arg other: The other point (L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance (C{meter}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit and/or if this and the B{C{other}} point are coincident or near-antipodal. @example: >>> p = LatLon(50.06632, -5.71475) >>> q = LatLon(58.64402, -3.07009) >>> d = p.distanceTo(q) # 969,954.166 m ''' return self._inverse(other, False, wrap).distance def distanceTo3(self, other, wrap=False): '''Compute the distance, the initial and final bearing along a geodesic between this and an other point, using Vincenty's inverse method. The distance is in the same units as this point's datum axes, conventially meter. The distance is measured on the surface of the ellipsoid, ignoring this point's height. The initial and final bearing (forward and reverse azimuth) are in compass C{degrees360} from North. @arg other: Destination point (L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: A L{Distance3Tuple}C{(distance, initial, final)}. @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit and/or if this and the B{C{other}} point are coincident or near-antipodal. ''' return self._xnamed(self._inverse(other, True, wrap)) @property_doc_(''' the convergence epsilon (C{scalar}).''') def epsilon(self): '''Get the convergence epsilon (C{scalar}). ''' return self._epsilon @epsilon.setter # PYCHOK setter! def epsilon(self, eps): '''Set the convergence epsilon. @arg eps: New epsilon (C{scalar}). @raise TypeError: Non-scalar B{C{eps}}. @raise ValueError: Out of bounds B{C{eps}}. ''' self._epsilon = Scalar_(eps, name='epsilon') def finalBearingOn(self, distance, bearing): '''Compute the final bearing (reverse azimuth) after having travelled for the given distance along a geodesic given by an initial bearing from this point, using Vincenty's direct method. See method L{destination2} for more details. @arg distance: Distance (C{meter}). @arg bearing: Initial bearing (compass C{degrees360}). @return: Final bearing (compass C{degrees360}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit. @example: >>> p = LatLon(-37.95103, 144.42487) >>> b = 306.86816 >>> f = p.finalBearingOn(54972.271, b) # 307.1736 ''' return self._direct(distance, bearing, False).final def finalBearingTo(self, other, wrap=False): '''Compute the final bearing (reverse azimuth) after having travelled along a geodesic from this point to an other point, using Vincenty's inverse method. See method L{distanceTo3} for more details. @arg other: The other point (L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Final bearing (compass C{degrees360}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit and/or if this and the B{C{other}} point are coincident or near-antipodal. @example: >>> p = new LatLon(50.06632, -5.71475) >>> q = new LatLon(58.64402, -3.07009) >>> f = p.finalBearingTo(q) # 11.2972° >>> p = LatLon(52.205, 0.119) >>> q = LatLon(48.857, 2.351) >>> f = p.finalBearingTo(q) # 157.9 ''' return self._inverse(other, True, wrap).final def initialBearingTo(self, other, wrap=False): '''Compute the initial bearing (forward azimuth) to travel along a geodesic from this point to an other point, using Vincenty's inverse method. See method L{distanceTo3} for more details. @arg other: The other point (L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Initial bearing (compass C{degrees360}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit and/or if this and the B{C{other}} point are coincident or near-antipodal. @example: >>> p = LatLon(50.06632, -5.71475) >>> q = LatLon(58.64402, -3.07009) >>> b = p.initialBearingTo(q) # 9.141877° >>> p = LatLon(52.205, 0.119) >>> q = LatLon(48.857, 2.351) >>> b = p.initialBearingTo(q) # 156.11064° @JSname: I{bearingTo}. ''' return self._inverse(other, True, wrap).initial @property_RO def iteration(self): '''Get the iteration number (C{int} or C{0} if not available/applicable). ''' return self._iteration + 1 # index to number @property_doc_(''' the iteration limit (C{int}).''') def iterations(self): '''Get the iteration limit (C{int}). ''' return self._iterations @iterations.setter # PYCHOK setter! def iterations(self, limit): '''Set the iteration limit. @arg limit: New iteration limit (C{int}). @raise TypeError: Non-scalar B{C{limit}}. @raise ValueError: Out-of-bounds B{C{limit}}. ''' self._iterations = Number_(limit, name='limit', low=4, high=500) def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None '''Convert this point to C{Vincenty}-based cartesian (ECEF) coordinates. @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} and other keyword arguments, ignored if B{C{Cartesian=None}}. Use B{C{Cartesian=...}} to override this L{Cartesian} class or specify B{C{Cartesian=None}}. @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{Cartesian}}, B{C{datum}} or other B{C{Cartesian_datum_kwds}}. ''' kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum) return LatLonEllipsoidalBase.toCartesian(self, **kwds) def _direct(self, distance, bearing, llr, height=None): '''(INTERNAL) Direct Vincenty method. @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit. ''' E = self.ellipsoid() c1, s1, t1 = _r3(self.lat, E.f) i = radians(bearing) # initial bearing (forward azimuth) si, ci = sincos2(i) s12 = atan2(t1, ci) * 2 sa = c1 * si c2a = 1 - sa**2 if c2a < EPS: c2a = 0 A, B = 1, 0 else: # e22 == (a / b)**2 - 1 A, B = _p2(c2a * E.e22) s = d = distance / (E.b * A) for self._iteration in range(self._iterations): ss, cs = sincos2(s) c2sm = cos(s12 + s) s_, s = s, d + _ds(B, cs, ss, c2sm) if abs(s - s_) < self._epsilon: break else: raise VincentyError(_no_convergence_, txt=repr(self)) # self.toRepr() t = s1 * ss - c1 * cs * ci # final bearing (reverse azimuth +/- 180) r = degrees360(atan2(sa, -t)) if llr: # destination latitude in [-90, 90) a = degrees90(atan2(s1 * cs + c1 * ss * ci, (1 - E.f) * hypot(sa, t))) # destination longitude in [-180, 180) b = degrees180(atan2(ss * si, c1 * cs - s1 * ss * ci) - _dl(E.f, c2a, sa, s, cs, ss, c2sm) + radians(self.lon)) h = self.height if height is None else height d = self.classof(a, b, height=h, datum=self.datum) else: d = None return Destination2Tuple(d, r) def _inverse(self, other, azis, wrap): '''(INTERNAL) Inverse Vincenty method. @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: If this and the B{C{other}} point's L{Datum} ellipsoids are not compatible. @raise VincentyError: Vincenty fails to converge for the current L{LatLon.epsilon} and L{LatLon.iterations} limit and/or if this and the B{C{other}} point are coincident or near-antipodal. ''' E = self.ellipsoids(other) c1, s1, _ = _r3(self.lat, E.f) c2, s2, _ = _r3(other.lat, E.f) c1c2, s1c2 = c1 * c2, s1 * c2 c1s2, s1s2 = c1 * s2, s1 * s2 dl, _ = unroll180(self.lon, other.lon, wrap=wrap) ll = dl = radians(dl) for self._iteration in range(self._iterations): ll_ = ll sll, cll = sincos2(ll) ss = hypot(c2 * sll, c1s2 - s1c2 * cll) if ss < EPS: # coincident or antipodal, ... if self.isantipodeTo(other, eps=self._epsilon): t = '%r %sto %r' % (self, _antipodal_, other) raise VincentyError(_ambiguous_, txt=t) # return zeros like Karney, but unlike Veness return Distance3Tuple(0.0, 0, 0) cs = s1s2 + c1c2 * cll s = atan2(ss, cs) sa = c1c2 * sll / ss c2a = 1 - sa**2 if abs(c2a) < EPS: c2a = 0 # equatorial line ll = dl + E.f * sa * s else: c2sm = cs - 2 * s1s2 / c2a ll = dl + _dl(E.f, c2a, sa, s, cs, ss, c2sm) if abs(ll - ll_) < self._epsilon: break # # omitted and applied only after failure to converge below, see footnote # # under Inverse at <https://WikiPedia.org/wiki/Vincenty's_formulae> # # <https://GitHub.com/ChrisVeness/geodesy/blob/master/latlon-vincenty.js> # elif abs(ll) > PI and self.isantipodeTo(other, eps=self._epsilon): # raise VincentyError('%s, %r %sto %r' % ('ambiguous', self, # _antipodal_, other)) else: t = _antipodal_ if self.isantipodeTo(other, eps=self._epsilon) else NN raise VincentyError(_no_convergence_, txt='%r %sto %r' % (self, t, other)) if c2a: # e22 == (a / b)**2 - 1 A, B = _p2(c2a * E.e22) s = A * (s - _ds(B, cs, ss, c2sm)) b = E.b # if self.height or other.height: # b += self._havg(other) d = b * s if azis: # forward and reverse azimuth sll, cll = sincos2(ll) f = degrees360(atan2(c2 * sll, c1s2 - s1c2 * cll)) r = degrees360(atan2(c1 * sll, -s1c2 + c1s2 * cll)) else: f = r = 0 return Distance3Tuple(d, f, r) def _dl(f, c2a, sa, s, cs, ss, c2sm): '''(INTERNAL) Dl. ''' C = f / 16.0 * c2a * (4 + f * (4 - 3 * c2a)) return (1 - C) * f * sa * (s + C * ss * (c2sm + C * cs * (2 * c2sm**2 - 1))) def _ds(B, cs, ss, c2sm): '''(INTERNAL) Ds. ''' c2sm2 = 2 * c2sm**2 - 1 ss2 = (4 * ss**2 - 3) * (2 * c2sm2 - 1) return B * ss * (c2sm + B / 4.0 * (c2sm2 * cs - B / 6.0 * c2sm * ss2)) def _p2(u2): # e'2 WGS84 = 0.00673949674227643 '''(INTERNAL) Compute A, B polynomials. ''' A = fpolynomial(u2, 16384, 4096, -768, 320, -175) / 16384 B = fpolynomial(u2, 0, 256, -128, 74, -47) / 1024 return A, B def _r3(a, f): '''(INTERNAL) Reduced cos, sin, tan. ''' t = (1 - f) * tan(radians(a)) c = 1 / hypot1(t) s = t * c return c, s, t def areaOf(points, datum=Datums.WGS84, wrap=True): # PYCHOK no cover '''DEPRECATED, use function C{ellipsoidalKarney.areaOf}. ''' from pygeodesy.ellipsoidalKarney import areaOf return areaOf(points, datum=datum, wrap=wrap) def intersections2(center1, rad1, center2, rad2, height=None, wrap=False, equidistant=None, tol=_TOL_M, LatLon=LatLon, **LatLon_kwds): '''Iteratively compute the intersection points of two circles each defined by an (ellipsoidal) center point and radius. @arg center1: Center of the first circle (L{LatLon}). @arg rad1: Radius of the second circle (C{meter}). @arg center2: Center of the second circle (L{LatLon}). @arg rad2: Radius of the second circle (C{meter}). @kwarg height: Optional height for the intersection points, overriding the "radical height" at the "radical line" between both centers (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg equidistant: An azimuthal equidistant projection class (L{EquidistantKarney} or L{equidistant}) or C{None} for L{Equidistant}. @kwarg tol: Convergence tolerance (C{meter}). @kwarg LatLon: Optional class to return the intersection points (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if B{C{LatLon=None}}. @return: 2-Tuple of the intersection points, each a B{C{LatLon}} instance or L{LatLon4Tuple}C{(lat, lon, height, datum)} if B{C{LatLon}} is C{None}. For abutting circles, the intersection points are the same instance. @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting circles or no convergence for the B{C{tol}}. @raise TypeError: If B{C{center1}} or B{C{center2}} not ellipsoidal. @raise UnitError: Invalid B{C{rad1}}, B{C{rad2}} or B{C{height}}. @see: U{The B{ellipsoidal} case<https://GIS.StackExchange.com/questions/48937/ calculating-intersection-of-two-circles>}, U{Karney's paper <https://arxiv.org/pdf/1102.1215.pdf>}, pp 20-21, section 14 I{Maritime Boundaries}, U{circle-circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>} and U{sphere-sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} intersections. ''' from pygeodesy.azimuthal import Equidistant E = Equidistant if equidistant is None else equidistant return _intersect2(center1, rad1, center2, rad2, height=height, wrap=wrap, equidistant=E, tol=tol, LatLon=LatLon, **LatLon_kwds) def perimeterOf(points, closed=False, datum=Datums.WGS84, wrap=True): # PYCHOK no cover '''DEPRECATED, use function C{ellipsoidalKarney.perimeterOf}. ''' from pygeodesy.ellipsoidalKarney import perimeterOf return perimeterOf(points, closed=closed, datum=datum, wrap=wrap) __all__ += _ALL_OTHER(Cartesian, LatLon, intersections2, ispolar) # from .points # **) MIT License # # Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. 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